A bifurcation problem
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5z19-6z15z2+z1z24+2z1z3 |
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-2z16z2+2z12z23+2z2z3 |
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z12+z22-0.265625 |
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This polynomial system arises from a test for Numerical Bifurcation and it has
been extracted from [1]. This exemple is also in the PoSSo test suite.
Characteristics:
- There are 8 real solutions, which are in the box [-0.6,6]2×
[-5,5]:
| z1 |
z2 |
z3 |
| -.5153882032 |
.1185133007 |
-.01244559884 |
| .5015771103 |
.1185133007 |
.01238951314 |
| .2619366407 |
.4438628124 |
-.01319432095 |
| -.2619366407 |
.4438628124 |
-.01319432095 |
| -.5015771103 |
.0 |
.01238951314 |
| .5153882032 |
.0 |
-.01244559884 |
| .0 |
-.5153882032 |
.0 |
| .0 |
.5153882032 |
.0 |
- It can be splitted into the subsystems:
| {64 |
|
2-17,z |
|
,33554432 z |
|
+417605 } |
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| 83886080 |
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8-122683392 |
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6
-100663296 |
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5+112582656 |
|
4 |
|
| +53477376 |
|
3-22304000 |
|
2-7102464 z |
|
+1046469, |
|
|
|
| 262144 z |
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+ 262144 |
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6-471040 |
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4+125120 |
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2-4913
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- The number of complex roots is therefore 20.
Example 1:
var := [z[1],z[2],z[3]];
[5*z[1]^9-6*z[1]^5*z[2]+z[1]*z[2]^4+2*z[1]*z[3],
-2*z[1]^6*z[2]+2*z[1]^2*z[2]^3+2*z[2]*z[3], z[1]^2+z[2]^2-0.265625];
· Solution by J.P. Merlet
Methode: bisection avec analyse par intervalle utilisant
jacobienne et hessienne des equations.
Temps de calcul:1.46s sur SUN ULTRA 1
· Solution by B. Mourrain
In maple, suing the function grobner[gsolve], one obtain the
decomposition of the system, and solve it by univariate polynomial solving
and substitution.
With the packadge
multires
hiding one variable, we obtain the different coordinates of the roots.
See the maple session.
References
- [1]
-
R.B. Kearfott.
Some tests of generalized bisection.
ACM Transactions on Mathematical Software, 13(3):197--220,
1987.
This document was translated from LATEX by HEVEA.